reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;
reserve U1,U2,U for Universe;
reserve u,v for Element of U;

theorem Th62:
  card Rank omega = card omega
proof
  deffunc h(Ordinal) = Rank $1;
  consider L such that
A1: dom L = omega & for A st A in omega holds L.A = h(A) from ORDINAL2:
  sch 2;
  now
    let X,Y;
    assume X in rng L;
    then consider x being object such that
A2: x in dom L and
A3: X = L.x by FUNCT_1:def 3;
    assume Y in rng L;
    then consider y being object such that
A4: y in dom L and
A5: Y = L.y by FUNCT_1:def 3;
    reconsider x,y as Ordinal by A2,A4;
A6: Y = Rank y by A1,A4,A5;
A7: x c= y or y c= x;
    X = Rank x by A1,A2,A3;
    then X c= Y or Y c= X by A6,A7,CLASSES1:37;
    hence X,Y are_c=-comparable;
  end;
  then
A8: rng L is c=-linear;
A9: card omega c= card Rank omega by CARD_1:11,CLASSES1:38;
A10: now
    let X;
    assume X in rng L;
    then consider x being object such that
A11: x in dom L and
A12: X = L.x by FUNCT_1:def 3;
    reconsider x as Ordinal by A11;
    X = Rank x by A1,A11,A12;
    hence card X in card omega by A1,A11,CARD_2:67,CARD_3:42;
  end;
  Rank omega = Union L by A1,Lm5,Th24
    .= union rng L by CARD_3:def 4;
  then card Rank omega c= card omega by A8,A10,CARD_3:46;
  hence thesis by A9;
end;
